Betaseries Correlations

Betaseries correlations track the trial-to-trial hemodynamic fluctuations that occur within a task. In contrast to typical General Linear Models (GLMs) where the hemodynamic fluctuations between trials of the same type are considered noise, betaseries correlations leverage these fluctuations to provide a quantitative measure of how synchronous they are across different brain regions identifying networks that are activated during a task. GLMs, by contrast, can only provide information about which brain regions are active, but cannot tell us which of those active regions form a network. For example, in a task where you have encode both the size and position of a stimulus, a GLM will show you active regions that may be encoding size or position or both. Betaseries correlations, on the other hand, could provide evidence of which regions form a size encoding network, and which form a position encoding network.

Conceptual Background

Jesse Rissman [Rissman, 2004]_ was the first to publish on betaseries correlations describing their usage in the context of a working memory task. In this task, participants saw a cue, a delay, and a probe, all occurring close in time. Specifically, a cue was presented for one second, a delay occurred for seven seconds, and a probe was presented for one second. Given the hemodynamic response function takes approximately six seconds to reach its peak, and generally takes over 20 seconds to completely resolve, we can begin to see a problem. The events within the trials occur too close to each other to discern what brain regions are related to encoding the cue, the delay, or the probe. To discern how the activated brain regions form networks, Rissman conceptualized betaseries correlations. In brief, instead of having a single regressor to describe all the cue events, a single regressor for all the delay events, and a single regressor for all the probe events, there is an individual regressor for every event in the experiment. For example, if your experiment has 40 trials, each with a cue, delay, and probe event, the GLM will have a total of 120 regressors, fitting a beta estimate for each trial. Complete this process for each trial of a given event type (e.g. cue), at the end you will have 4D volume where each volume represents the beta estimates for a particular trial, and each voxel represents a specific beta estimate.

Having one regressor per trial in a single model is known as least squares all. This method, however, has limitations in the context of fast event related designs. Since each trial has its own regressor, trials that occur very close in time are colinear (remember the hemodynamic response is slow). Jeanette Mumford et al. [Mumford, 2012]_ investigated this issue in the context of image classification. In this article, Mumford introduces another modelling strategy known as least squares separate. In this modelling strategy, instead of having one GLM with a regressor per trial, least squares separate implements a GLM per trial with two regressors: 1) one for the trial of interest, and 2) one for every other trial in the experiment. This process reduces the colinearity of the regressors and creates a more valid estimate of how each regressors fits the data.

Math Background

\[\begin{equation} Y = X\beta + \epsilon \end{equation}\]

The above equation is the General Linear Model (GLM) presented using matrix notation. \(Y\) represents the time-series we are attempting to explain. The \(\beta\) assumes any value that minimizes the squared error between the modeled data and the actual data, \(Y\). Finally, \(\epsilon\) (epsilon) refers to the error that is not captured by the model. Within a GLM, trial-to-trial betas may be averaged for a given trial type and the variance is treated as noise. However, those trial-to-trial fluctuations may also contain important information that the typical GLM will ignore/penalize. With a couple modifications to the above equation, we arrive at calculating a betaseries.

\[\begin{equation} Y = X_1\beta_1 + X_2\beta_2 + . . . + X_n\beta_n + \epsilon \end{equation}\]

With the betaseries equation, a beta is estimated for every trial, instead of for each trial type (or whichever logical grouping). This gives us the ability to align all the trial betas from a single trial type into a list (i.e. series). This operation is completed for all voxels, giving us as many lists of betas as there are voxels in the data. Essentially, we are given a 4-D dataset where the fourth dimension represents trial number instead of time (as the fourth dimension is represented in resting state). And analogous to resting state data, we can perform correlations between the voxels to discern which voxels (or which aggregation of voxels) synchronize best with other voxels.

There is one final concept to cover in order to understand how the betas are estimated in NiBetaSeries. You can model individual betas using a couple different strategies; with a least squares all (LSA) approximation which the equation above represents, and a least squares separate (LSS) approximation in which each trial is given it’s own GLM. The advantage of LSS comes from reducing the colinearity between closely spaced trials. In LSA, if trials occurred close in time then it would be difficult to model whether the fluctuations should be attributed to one trial or the other. LSS reduces this ambiguity by only having two regressors: one for the trial of interest and another for every other trial. This reduces the colinearity between regressors and makes each beta estimate more reliable.

for trial, beta in zip(trials, betas):
    data = X[trial] * beta + error

This python psuedocode demonstrates LSS where each trial is given it’s own model.